Sala: Auditório Jacy Monteiro
Palestrante: Friedrich Martin Schneider (UFSC)
Título: Equivariant dissipation in permutation groups.
Resumo: In his seminal work on metric measure geometry, Gromov introduced the
observable distance, a metric on the set of isomorphism classes of
metric measure spaces. This metric generalizes the well-known measure
concentration phenomenon in a very natural way: a sequence of spaces has
the Levy concentration property if and only if it converges to a
singleton space with respect to the observable distance. Motivated by
striking applications of measure concentration in topological dynamics,
Vladimir Pestov proposed to study the phenomenon of concentration to
non-trivial spaces in the context of topological groups. Prompted by
work of Glasner and Weiss, in 2006 he posed the following problem: given
a left-invariant metric d on the full symmetric group of the natural
numbers, compatible with the topology of pointwise convergence, does the
sequence of finite symmetric groups, equipped with their normalized
counting measures and the restrictions of d, concentrate to the compact
space of linear order on the natural numbers, endowed with its unique
invariant Borel probability measure and a suitable compatible metric?
In the talk, I will answer Vladimir Pestov's question in the negative. As we will see, the above sequence of finite symmetric groups indeed dissipates (in the sense of Gromov), thus does not even admit a subsequence being Cauchy with respect to the observable distance.
In the talk, I will answer Vladimir Pestov's question in the negative. As we will see, the above sequence of finite symmetric groups indeed dissipates (in the sense of Gromov), thus does not even admit a subsequence being Cauchy with respect to the observable distance.